ferred to here follow from theorems 3 and 4 of [2]. ... are closed linear operators from X to Y, D(A) is dense in X, D{B) 2 £>(A), .... Xx and Fx satisfy all the conclusions of the theorem. To conclude the proof, it suffices to produce complementary ...
PACIFIC JOURNAL OF MATHEMATICS Vol. 15, No. 1, 1965
DECOMPOSITION THEOREMS FOR FREDHOLM OPERATORS T. W.
GAMELIN
This paper is devoted to proving and discussing several consequences of the following decomposition theorem: Let A and B be closed densely-defined linear operators from the Banach space X to the Banach space Y such that D(B) 2 D(A), D(B*) 2 JD(A*), the range R(A) of A is closed, and the dimension of the null-space N(A) of A is finite. Then X and Y can be decomposed into direct sums X = Xo Θ Xu Y = γ0 0 γl9 where Xί and Yx are finite dimensional, Xλ g D(A), XonD(A) is dense in X, and (Xo, Yo) and (Xί9 Yi) are invariant pairs of subspaces for both A and 5. Let A* and Bi be the restrictions of A and J5 respectively to Xu For all integers ft, (tfoAΓXO) £ #(A), and dim (BoAϊ^CO) =fc dim (EoA^XO) = ft dim #(A 0 ) . Also, the action of Ai and Bι from Xi to Yi can be given a certain canonical description. The object of this paper is to study the operator equation Ax — XBx =y, where A and B are (unbounded) linear operators from a Banach space X to a Banach space Y. In §1, an integer μ(A:B) is defined, which expresses a certain interrelationship between the null space of A and the null space of B. In §1 and 2, decomposition theorems are proved which refine theorem 4 of [2]. The theorems allow us to split off certain finite dimensional invariant pairs of subspaces of X and Y so that A and B are well-behaved with respect to μ(A:B) on the remainder. In §4, the stability of these decompositions under perturbation of A by XB is investigated. In § 5, relations between the dimensions of certain subspaces of X and Y are given, and a formula for the Fredholm index of A — XB is obtained. These extend results of Kaniel and Schechter [1], who consider the case X= Y and B the indentity operator. It should be noted that the results of Kaniel and Schechter referred to here follow from theorems 3 and 4 of [2]. The results of this paper properly refine Kato's results only when the null space of B is not {0}. 1. We will be considering linear operators T defined on a dense linear subset D(A) of a Banach space X9 and with values in a Banach space Y. N(T) and R(T) will denote the null space and range of T respectively, while a{T) is the dimension of N{T), and β(T) is the Received February 10, 1964. 97
T. W. GAMELIN
codimension of R(T) in Y.
T is a Fredholm operator if T is closed,
R(T) is closed, and both a{T) and β(T) are finite. The index of a Fredholm operator is the integer.
κ(T) = a{T) - β{T) . Let P be a subspace of X, Q a subspace of Y. (P, Q) is an invariant pair of subspaces for T if Γ(P Π J5(Γ)) S Q Standing assumptions: In the remainder of the paper, A and I? are closed linear operators from X to Y, D(A) is dense in X, D{B) 2 £>(A), and ΰ ( β * ) g ΰ ( A * ) ; A is semi-Fredholm, in the sense that R(A) is closed and α(A)< co. The assumption D(B*) Ξ2 £)(A*) seems necessary for the proof of the decomposition theorems. It is often met when A and B are differential operators on some domain in Euclidean space, and the order of B is less than the order of A. It is always met when B is bounded. The linear manifolds Nk = Nk(A:B) and Mk = Mk(A:B) are defined by induction as follows :
N, = N(A) Nk = A-XBN^) Mk = BNk .
,
k>1
Nk and Mk are increasing sequences of linear manifolds in X and Y respectively. The smallest integer n such that Nn is not a subset of U^ϋ^A) will be denoted by v(A:B). If Nn is a subset of B~λR{A) for all n, then we define v(A:B) = co. (cf. [2]) The dimension of Nk will be denoted by πk — πk{A:B), and the dimension of Mk by pk — pk(A:B). Then ^ = a(A), and, in general, πk^ka(A). μ(A:B) will denote the first integer n such that πn< na(A). It Vrn = na(A) for all intergers n, then we define μ{A: B) = co. In general, μ(A: 1?) ^ v(A:B) + 1. This inequality is trivial if i£= oo. If v < co, then Λf v -iSi2(A), while Mv g i?(A). Consequently, 7ΓV+1 < 7ΓV + α(A) ^ (v + l)a(A), and so /i(A: B) ^ y + 1. We define 0"fc(A :B) = πk — ττA;_1. Then σfc is the dimension of the quotient space N^N^. {σk} is a decreasing sequence of nonnegative integers, and so the limit σ(A : β) — lim σk(A : B)
exists.
If μ(A : B) = co, then σ(A : B) = α(A). 2* THEOREM 1. Assume, in addition to the standing assumptions on A and B, that v{A: B) = co. TΛen X and Y" can be decomposed
DECOMPOSITION THEOREMS FOR FREDHOLM OPERATORS
99
into direct sums
Z=Z o 0X 1
Y= r , φ r 1 ( where Xλ and Yx are finite dimensional, X1 S D(A), Xo 0 D(A) is dense in Xo, and (Xo, Yo) and (X19 Yi) are invariant pairs for both A and B. If A{ and B{ are the restrictions of A and B respectively to Xi9 then μ(A0, Bo) = oo, while Aλ and Bλ map Xx onto Yx. Furthermore, X1 and Yλ can be decomposed as direct sums
where Ax and Bx map P3 onto Q3 . Bases {x)\ 1 < i ^ η{j)} and {y)\ 1 5g i ^ Ύ](j) — 1} can be chosen for P3 and Qo- respectively so that - Bx) = yj, Ax) = 0 = Bxfj) . γ
l ^ i
Although the decomposition is not, in general, unique, the integers φ and η(j), 1 ^ j ^ m, are uniquely determined by A and J5. In fact, p = α(A) - (τ(A: B) . Proof. Let n — a{A), and suppose that {z\, zl} is a basis for N(A). Since v(A: B) = oo, z) can be chosen by induction so that Az) — Bz)~x. \z\\ 1 ^ j S n, 1 ^ i ^ m} is a spanning set for Nm, while {β^}: 1 ^ j ^ n, 1 ^ i ^ m} is a spanning set for Λfw. Also, {z?: 1 ^ i ^ n} span iVm modulo ^ ^ Recall that σm = σ(m) = dim (iVm/iVm_1). By induction, the order of the 2} can be chosen so that {2?_σ(m)+i, , C} span ΛΓm modulo Nn-^ Then Gm = {z}: n - σ(ΐ) + l ^ i ^ ^ ,
1 ^
is a basis for i\Γm. Let ^(i) be the greatest integer k such that z) e Gk. If z) e for all &, let ^(jΓ) = oo. Then 1 S η(l) ^ η(2) ^ S η{n). Let p be the greatest integer k such that Ύ](k) < oo. By definition of σ, it is clear that p — α(A) — a . Suppose 1 ^ j ^ p. and so we can write
2j°'
)+1
is linearly dependent on the set Gη{j)+1,
100
T. W. GAMELIN
where the sum is taken over all pairs of integers (i, k), with t h e understanding t h a t z\ — 0 if ΐ ^ O and α α = 0 if z\ $ Gvij)+1. For — 1 ^ g ^ η(j) define
For 0 ^
?
^ τ?(i),
)—Q+1
V
In particular, Bxfj) = 0. Since the sum for ^ ( i ) ~ 9 involves zfj)~q only in the first term, the z]{j)-q may be replaced by the xf3Ί~qf 0 ^ q ^ η(j), to obtain another basis for Nη{j)+1. Repeating this process for 1 ^ j g>p, and making other appropriate replacements, we arrive at vectors x) such that. (1)
x\, •••,«* are a basis for N(A)
(2)
Bx) = Ax}+1 ,
1^ ΐ S
B^(i) = 0 ,
(3)
l^
For convenience, it is assumed that (4)
x) = 0
if i >
If l ^ i ^ P , let P y be the subspace of X with basis {x}, -,xfj)}. Let Qj be the subspace of Y with basis {y)9 •• , ^ ( i ) " 1 } , where yj — Bx) - i4α?J+1. Let Xx = Pλ 0 0 Pp and ^ = Qx © © Qp. Then Xx and F x satisfy all the conclusions of the theorem. To conclude the proof, it suffices to produce complementary subspaces to Xx and Yx which also form an invariant pair. We will construct functionals {g): 1 ^ i ^ 7}(j) ,
H i ^ r f o n l
and
{/}: 1 ^ i ^ 5?(i) - 1 , U i ^ r f o n Γ such that the fj are in the domain of A* and (5)
0}+α = A*S\ ,
(6)
9) = B*f} ,
(7)
ΓM)
= δitδjk ,
l g ί g V{j) ~ 1 l^iS
V(j) - 1
1 S j, k g n
DECOMPOSITION THEOREMS FOR FREDHOLM OPERATORS
(8)
g){xϊ) = δjjk
,
101
1^ j, k ^ n H g _ i Ξ> 1, to satisfy (5) +1 ι +1 through (8). By (8), gi is orthogonal to N(A), and so g k is in the closure of R(A*). Since R{A) is closed, R(A*) is closed, and there is +1 an fi e D(A*) for which A*f{ - gi . Let g\ = B*f{. Then (5) and (6) hold by definition. To verify (7), we have for q ^ ί, +1
fXvl) = f}(Axl ) = (A*/J)(aJi +1 ) = 9)+1(xl+1)
= δiqδSk
.
(8) is an immediate consequence of (7). Let Xo = n {N(g}): l ^ i ^
vU)
,
^o = ΓΊ {ΛΓ(/}): 1 (A) n Xo Then /j(Aα ) = (A*/j)(a) = g)+1(x) = 0, and so Ax e Y"o Similarly, Bx G YO, and (Xo, YQ) is an invariant pair for both A and J5. Since (X09 Yo) and (X1? Yλ) are invariant pairs, JVA(i4. : S ) i l I 0 = ΛΓ,(Λ : B0)β For k sufficiently large, Xx g iVA(A : B), and so dim {iV/cfl(Λ : B,)INk(A,: 50)} - dim {ΛΓ/C+1(A : B)/Nk(A : =σ = a(A)
-
v
This can occur only if dim Nk(A0: Bo) = &α:(A0) for all integers k. Hence ^(Ao: Bo) = oo. 3* Let (P, Q) be an invariant pair of finite dimensional subspaces for A and B. (P, Q) is an irreducible invariant pair of type v if there are bases {#J?=1 for P and {!/
